i edited post

but i think there is mistake there (probably typo) with those solutions, because okay (you don't need to) like put variable "t" to substitute "z" lol unknown as unknown

so they expressed all solutions (and here is mistake) for x,y,w (not two different x) don't see how else did they lost the "w"

and to describe family of solutions, you must know how do depending on those solutions (for let's say t-2=0 -> t= 2) how does that effect the system of equations

do you know that? (solutions of the system with not homogeneous equations

)

quick reminder if you forget

1° when $\displaystyle D\neq 0 $ and if at least one of $\displaystyle D_x\vee D_y\vee D_z\vee \neq 0$ than solutions are unique and given

$\displaystyle \displaystyle x = \frac {D_x}{D} , y= \frac {D_y}{D} , z = \frac {D_z}{D}$

2° if $\displaystyle D= 0 $ and if at least one of $\displaystyle D_x\vee D_y\vee D_z\vee \neq 0$ than system of equations have no solutions

3° if $\displaystyle D= 0 $ and if $\displaystyle D_x= D_y= D_z= 0$ than there can be :

a) all sub determinants of D are zeros, and coefficients with the unknowns are proportional there are 2 cases :

- all independent members of system are proportional with coefficient with unknowns than there are infinity of solutions

- all independent members of system are not proportional with coefficient with unknowns than there are no solutions

b) if at least one sub determinant of D are different from zero than there are infinity of solution

Edit: or if this is something that you still didn't learn in school , than you can do it like this :

1° $\displaystyle x+y+z+w = 4$

2° $\displaystyle x+3y+3z =2$

3° [tex] x+y+2z-w=6[tex]

than just add first and third equations so that you will get rid of "w"

1°+3° ->

1°+3° $\displaystyle 2x+2x+3z =10$

2° $\displaystyle x+3y+3z = 2$

and now you again have three unknowns and just two equations

so there is again infinity of solution (and now that t=z have meaning) so you get :

$\displaystyle 2x+2y=10-3t$

$\displaystyle x+3y=2-3t $

multiply second with (-2) and add to the first equation

$\displaystyle 2x+2y=10-3t$

$\displaystyle -2x-6y=6t-4 $

1°+ 2° -> $\displaystyle -4y=6+3t Rightarrow y= \frac {-6-3t}{4} = \frac {-3(2+t)}{4} $ and so on